Optimal. Leaf size=189 \[ \frac{(a+b x) e^{-\frac{A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-\frac{m+1}{n}} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^p \left (-\frac{(m+1) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{B n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)} \]
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Rubi [F] time = 0.975122, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (210 c+210 d x)^{-2-m} (a g+b g x)^m \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^p \, dx &=\int (210 c+210 d x)^{-2-m} (a g+b g x)^m \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^p \, dx\\ \end{align*}
Mathematica [F] time = 0.537676, size = 0, normalized size = 0. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 3.169, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{m} \left ( dix+ci \right ) ^{-2-m} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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